Properties

Label 1.139.s
Base field $\F_{139}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{139}$
Dimension:  $1$
L-polynomial:  $1 + 18 x + 139 x^{2}$
Frobenius angles:  $\pm0.776456669103$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-58}) \)
Galois group:  $C_2$
Jacobians:  $2$

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $158$ $19276$ $2683946$ $373337568$ $51888419918$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $158$ $19276$ $2683946$ $373337568$ $51888419918$ $7212551982124$ $1002544381232762$ $139353666624137088$ $19370159751220200254$ $2692452204120278342476$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{139}$.

Endomorphism algebra over $\F_{139}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-58}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
1.139.as$2$(not in LMFDB)